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# Universal property of the product

Theorem : The universal property of the product for sets is equivalent to the Cartesian product of those sets.

Proof : The universal property of the product is the following : For a category $C$, let $X_1$ and $X_2$ be two objects of $C$. The product $X_1 \times X_2$ is an object of $C$ equipped with the morphisms $\pi_i : X_1 \times X_2 \to X_i$, $i = 1, 2$, satisfying the universal property that, for every object $Y$ and every pair of morphisms $f_i : Y \to X_i$, there exists a unique morphism $f : X_1 \times X_2$ such that